RationalSolutions

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\centerline{\fbox{\parbox{10cm}{\Large\bf UserFunction RationalSolutions} }}
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\vspace*{5mm}\noindent{\bf RationalSolutions}$(u:LODE(RATF Q,y,x))$
The single argument $u$ is a linear ordinary differential equation for the unknown function $y$. Its  coefficients are rational functions in the independent variable $x$. The meaning of the parameters is as follows.


$C$: Coefficient type.

$y$: Dependent variable.

$y$: Independent variable.

\vspace*{1mm}\noindent{\bf Specification.}
The rational solutions of $u$ are returned.


\vspace*{1mm}\noindent{\bf Examples.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window, $D=d/dx$.


\4{\tt deq:=Df(y,x,3)+8*x/(x**2+1)*Df(y,x)+10/(x**2+1)*Df(y,x)=0;}

\4{\tt rs:=RationalSolutions deq;}


\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{

The Differential Equation:

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${y'''}+\fracsm{8x}{x^{2}+1}{y''}+\fracsm{10}{x^{2}+1}{y'} = 0$

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Fundamental system of rational solutions:

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$y=\{1,\1\fracsm{1}{x^{4}+2x^{2}+1},\1\fracsm{\Frac{1}{2}x^{3}+ \Frac{3}{2}x}{x^{4}+2x^{2}+1}\}$  }}

\vspace*{5mm}\noindent{\bf RationalSolutions}$(u:RICCATI(RATF Q,y,x))$
The single argument $u$ is a Riccati equation for the unknown function $y$. Its  coefficients are rational functions in the independent variable $x$. The meaning of the parameters as above.

\vspace*{1mm}\noindent{\bf Specification.}
The rational solutions of $u$ are returned.


\vspace*{1mm}\noindent{\bf Examples.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window, $D=d/dx$.


\4{\tt deq:=Df(q,x)+q**2-30/(x**2-10*x+25)=0;}

\4{\tt rs:=RationalSolutions deq;}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{

The Differential Equation:

$q'+q^2-\fracsm{30}{x^2-10x+25}=0$

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RationalSolutions:
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$q=-\Frac{\fracsm{3}{2}}{x-\fracsm{2}{3}}
    +\fracsm{324x^3-324x^2+108x-12}{C+81x^4-108x^3+54x^2-12x+1}$ }}

 

\vspace*{5mm}\noindent{\bf RationalSolutions}$(u:LDFMOD(RATF Q,dv,iv,O)$
The single argument $u$ is a linear ordinary differential equation for the unknown function $y$. Its  coefficients are rational functions in the independent variables.

dv: Dependent variables.

iv: Independent variables.


\vspace*{1mm}\noindent{\bf Example.}

\4{\tt sys:=\{df(z,x,2)+4/x*df(z,x)+2/x**2*z,df(z,x,y)+1/x*df(z,y),}

\hspace*{20mm}{\tt df(z,y,2)+1/y*df(z,y)-x/y**22*df(z,x)-2/y**2*z\};}

\4{\tt rs:=RationalSolutions sys;}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{

The Differential Equations:
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$z_{x,x}+\fracsm{4}{x}z_{x}+\fracsm{2}{x^{2}}z=0$
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$z_{x,y}+\fracsm{1}{x}z_{y}=0$
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$z_{y,y}+\fracsm{1}{y}z_{y}-\fracsm{x}{y^{2}}z_{x}-\fracsm{2}{y^{2}}z=0$

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Term Order: GRLEX, $z,y>x$

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Fundamental system:

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$z =\{\fracsm{y}{x},\1\fracsm{1}{xy},\1\fracsm{1}{x^{2}}\}$ }}

\vspace*{1mm}\noindent{\bf Related Functions:} PolynomialSolutions.


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{\bf References}

F.~Schwarz, {\em Algorithmic Lie Theory for Solving Ordinary Differential Equations}, CRC Press, 2007.


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